Modes in a non-neutral plasma of finite length, m=0,1

For realistic, cold equilibria of finite length representing a pure electron plasma confined in a cylindrical Malmberg–Penning trap, the mode spectrum for Trivelpiece–Gould, mϭ0, and for diocotron, mϭ1, modes is calculated numerically. A novel method involving finite elements is used to successfully compute eigenfrequencies and eigenfunctions for plasma equilibria shaped like pancakes, cigars, long cylinders, and all things in between. Mostly sharp-boundary density configurations are considered but also included in this study are diffuse density profiles including ones with peaks off axis leading to instabilities. In all cases the focus has been on elucidating the role of finite length in determining mode frequencies and shapes. For mϭ0 accurate eigenfrequencies are tabulated and their dependence on mode number and aspect ratio is computed. For mϭ1 it is found that the eigenfrequencies are 2% to 3% higher than given by the Fine–Driscoll formula ͓Phys. Plasmas 5, 601 ͑1998͔͒. The ''new modes'' of Hilsabeck and O'Neil ͓Phys. Plasmas 8, 407 ͑2001͔͒ are identified as Dubin modes. For hollow profiles finite length in cold-fluid can account for up to ϳ70% of the theoretical instability growth rate.